I got my Class A working for the City of Portland Urban Forestry division. I still have it even though I don't drive anything large anymore. The instructors, test proctors, supervisors, and operators told me that anything driven for commerce towing a trailer rated over 10k is Class A territory. Personally owned/used equipment might leave space for a looser interpretation, irrelevant if your business name is on the rig. Doesn't matter if it is a pickup or dump truck. The Class B clears you for vehicle weights over 26k, not trailer weights though.
Enforcement varies widely here Oregon though. I know plenty of contractors running without CDLs towing blatantly heavy trailers behind pickups and 1-ton trucks in my area, but they won't leave our county because they know they'll get popped. I would rather stay way under the thresholds and operate a smaller setup.
All that said, that is a sweet truck and I hope you sell it soon.